The Discrete Boltzmann Equation: The Regular Plane Model with Four Velocities

Abstract. For a simple discrete model of Boltzmann equation, we study the derivatives of H-Boltzmann function, and prove that all derivatives of odd order are negative, instead all derivatives of even order are postive. These result is a first and small generalisation of the classical H-Boltzmann theorem. We consider a discrete model of the Boltzmann equation with the the four following velocities: ( ) ( ) ( ) ()1,0,0,1,1,0,0,1 4321 −=−== = cccc ξξξξ. The only binary collisions are ( ) ()4231,, ξξξξ ↔. Denoting by)(tiN the density of molecules with velocity ξi, and by